Scalar and vector potentials in electromagnetism. The scalar potential is potential energy per unit charge. For potential energy, use the potential-energy tag.

In electromagnetic theory, there are two kinds of potential. First, and most common, is the scalar potential, denoted $\varphi$ or $\phi$, defined as the potential energy per unit charge of a charged object in an electric field.

$$\varphi(\vec{r}) = \frac{U(\vec{r})}{q}$$

Scalar potential should not be confused with the related concept of , denoted $V$, which is the difference between the scalar potential at two points.

In a static system, where charges and currents are constant (and thus the electric and magnetic fields are also constant), the is the gradient of scalar potential.

$$\vec{E} = -\vec{\nabla}\varphi$$

The other kind of potential is the vector potential, denoted $\vec{A}$, which is not directly related to potential energy but is related to the $\vec{B}$ via the curl:

$$\vec{B} = \vec{\nabla}\times\vec{A}$$

In , these two potentials are combined into a four-vector $A^{\mu}$, defined as

$$A^{\mu} = \biggl(\frac{\varphi}{c}, A_x, A_y, A_z\biggr)$$

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